High distance Heegaard splittings of 3-manifolds

Abstract

J. Hempel [J. Hempel, 3-manifolds as viewed from the curve complex, Topology 40 (3) (2001) 631–657] used the curve complex associated to the Heegaard surface of a splitting of a 3-manifold to study its complexity. He introduced the distance of a Heegaard splitting as the distance between two subsets of the curve complex associated to the handlebodies. Inspired by a construction of T. Kobayashi [T. Kobayashi, Casson–Gordon's rectangle condition of Heegaard diagrams and incompressible tori in 3-manifolds, Osaka J. Math. 25 (3) (1988) 553–573], J. Hempel [J. Hempel, 3-manifolds as viewed from the curve complex, Topology 40 (3) (2001) 631–657] proved the existence of arbitrarily high distance Heegaard splittings. In this work we explicitly define an infinite sequence of 3-manifolds { M n } via their representative Heegaard diagrams by iterating a 2-fold Dehn twist operator. Using purely combinatorial techniques we are able to prove that the distance of the Heegaard splitting of M n is at least n. Moreover, we show that π 1 ( M n ) surjects onto π 1 ( M n − 1 ) . Hence, if we assume that M 0 has nontrivial boundary then it follows that the first Betti number β 1 ( M n ) > 0 for all n ⩾ 1 . Therefore, the sequence { M n } consists of Haken 3-manifolds for n ⩾ 1 and hyperbolizable 3-manifolds for n ⩾ 3 .